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October 2017, 14(5&6): 1091-1117. doi: 10.3934/mbe.2017057

Effect of seasonal changing temperature on the growth of phytoplankton

1. 

School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin 130024, China

2. 

Department of Mathematics, Dalian Maritime University, 1 Linghai Road, Dalian, Liaoning, 116026, China

3. 

School of Urban and Environmental Sciences, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin 130024, China

4. 

Lamps and Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada

* Corresponding author: Meng Fan

Received  March 2016 Accepted  October 2016 Published  May 2017

Fund Project: Supported by NSFC-11671072 & 11271065, RFDP-20130043110001, RFCP-CMSP, and FRFCU-14ZZ1309

An non-autonomous nutrient-phytoplankton interacting model incorporating the effect of time-varying temperature is established. The impacts of temperature on metabolism of phytoplankton such as nutrient uptake, death rate, and nutrient releasing from particulate nutrient are investigated. The ecological reproductive index is formulated to present a threshold criteria and to characterize the dynamics of phytoplankton. The positive invariance, dissipativity, and the existence and stability of boundary and positive periodic solution are established. The analyses rely on the comparison principle, the coincidence degree theory and Lyapunov direct method. The effect of seasonal temperature and daily temperature on phytoplankton biomass are simulated numerically. Numerical simulation shows that the phytoplankton biomass is very robust to the variation of water temperature. The dynamics of the model and model predictions agree with the experimental data. Our model and analysis provide a possible explanation of triggering mechanism of phytoplankton blooms.

Citation: Ming Chen, Meng Fan, Xing Yuan, Huaiping Zhu. Effect of seasonal changing temperature on the growth of phytoplankton. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1091-1117. doi: 10.3934/mbe.2017057
References:
[1]

G. Ahlgren, Temperature functions in biology and their application to algal growth constants, Oikos, 49 (1987), 177-190. doi: 10.2307/3566025.

[2]

O. Bernard and B. Remond, Validation of a simple model accounting for light and temperature effect on microalgal growth, Bioresource Technology, 123 (2012), 520-527. doi: 10.1016/j.biortech.2012.07.022.

[3]

R. BouterfasM. Belkoura and A. Dauta, Light and temperature effects on the growth rate of three freshwater [2pt] algae isolated from a eutrophic lake, Hydrobiologia, 489 (2002), 207-217.

[4]

C. ButterwickS.I. Heaney and J. F. Talling, Diversity in the influence of temperature on the growth rates of freshwater algae, and its ecological relevance, Freshwater Biol, 50 (2005), 291-300. doi: 10.1111/j.1365-2427.2004.01317.x.

[5]

M. ChenM. FanR. LiuX. Yuan and H. P. Zhu, The dynamics of temperature and light on the growth of phytoplankton, J. Theor. Biol., 385 (2015), 8-19. doi: 10.1016/j.jtbi.2015.07.039.

[6]

W. Chen and A. Nauwerck, A note on composition and feeding of the crustacean zooplankton of Lake Taihu, Jiangsu Province, China, Limnologica, 26 (1996), 275-280.

[7]

Y. W. ChenB. Q. QinK. Teubner and M. T. Dokulil, Long-term dynamics of phytoplankton assemblages: Microcystis-domination in Lake Taihu, a large shallow lake in China, J. Plankton Res., 25 (2003), 445-453. doi: 10.1093/plankt/25.4.445.

[8]

L. DingY. Pang and L. Li, Simulation study on algal dynamics under different hydrodynamic conditions, Acta Ecologica Sinica, 25 (2005), 1863-1868.

[9]

X. H. DongH. BennionR. BattarbeeX. D. Yang and E. F. Liu, Tracking eutrophication in Taihu Lake using the diatom record: potential and problems, J. Paleolimnol, 40 (2008), 413-429. doi: 10.1007/s10933-007-9170-6.

[10]

P. Driessche and W. James, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[11]

J. J. ElserK. AcharyaM. KyleJ. CotnerW. MakinoT. MarkowT. WattsS. HobbieW. FaganJ. SchadeJ. Hood and R. W. Sterner, Growth rate-stoichiometry couplings in diverse biota, Ecology Letters, 6 (2003), 936-943. doi: 10.1046/j.1461-0248.2003.00518.x.

[12]

M. FanQ. Wang and X. Zou, Dynamics of a non-autonomous ratio-dependent predator-prey system, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 133 (2003), 97-118. doi: 10.1017/S0308210500002304.

[13]

P. J. S. Franks, NPZ models of plankton dynamics: Their construction, coupling to physics, and application, J. Oceanog, 58 (2002), 379-387.

[14]

J. A. FreundS. MieruchB. ScholzeK. Wiltshire and U. Feudel, Bloom dynamics in a seasonally forced phytoplankton zooooplankton model: trigger mechanisms and timing effects, Ecol. Complex, 3 (2006), 129-139.

[15]

G. F. FussmannS. P. EllnerK. W. Shertzer and N. G. Hairston, Crossing the hopf bifurcation in a live predator-prey system, Science, 290 (2000), 1358-1360. doi: 10.1126/science.290.5495.1358.

[16]

R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.

[17]

R. J. GeiderH. L. Maclntyre and T. M. Kana, A dynamic regulatory model of phytoplanktonic acclimation to light, nutrients, and temperature, Limnol. Oceanogr, 43 (1998), 679-694. doi: 10.4319/lo.1998.43.4.0679.

[18]

P. M. Glibert, Eutrophication and Harmful Algal Blooms: A Complex Global Issue, Examples from the Arabian Seas including Kuwait Bay, and an Introduction to the Global Ecology and Oceanography of Harmful Algal Blooms (GEOHAB) Programme, Int. J. Oceans and Oceanography, 2 (2007), 157-169.

[19]

J. C. Goldman and J. C. Edward, A kinetic approach to the effect of temperature on algal growth, Limnol. Oceanogr., 19 (1974), 756-766.

[20]

G. M. GrimaudV. L. GuennecS. D. AyataF. MariretA. Schiandra and O. Bernard, Modelling the effect of temperature on phytoplankton growth across the global ocean, IFAC-PapersOnLine, 48 (2015), 228-233. doi: 10.1016/j.ifacol.2015.05.059.

[21]

J. P. Grover and T. H. Chrzanowski, Seasonal dynamics of phytoplankton in two warm temperate reservoirs: association of taxonomic composition with temperature, J. Plankt. Res, 28 (2006), 1-17. doi: 10.1093/plankt/fbi095.

[22]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 5-60. doi: 10.4039/entm9745fv.

[23]

A. HuppertB. BlasiusR. Olinky and L. Stone, A model for seasonal phytoplankton blooms, J. Theor. Biol., 236 (2005), 276-290. doi: 10.1016/j.jtbi.2005.03.012.

[24]

K. S. JohnsonF. P. Chavez and G. E. Friederich, Continental-shelf sediment as a primary source of iron for coastal phytoplankton, Nature, 398 (1999), 697-700.

[25]

R. I. Jones, The importance of temperature conditioning to the respiration of natural phytoplankton communities, British Phycological Journal, 12 (2007), 277-285. doi: 10.1080/00071617700650291.

[26]

S. E. Jorgensen and G. Bendoricchio, Fundamentals of Ecological Modelling, Elsevier, 2001.

[27]

I. LoladzeY. Kuang and J. J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bulletin of Mathematical Biology, 62 (2000), 1137-1162. doi: 10.1006/bulm.2000.0201.

[28]

J. H. Luo, Phytoplankton-zooplankton dynamics in periodic environments taking into account eutrophication, Math. Biosci., 245 (2013), 126-136. doi: 10.1016/j.mbs.2013.06.002.

[29]

J. R. MoisanT. A. Moisan and M. R. Abbott, Modelling the effect of temperature on the maximum growth rates of phytoplankton populations, Ecol. Model., 153 (2002), 197-215. doi: 10.1016/S0304-3800(02)00008-X.

[30]

G. PhillipsR. JacksonC. Bennett and A. Chilvers, The importance of sediment phosphorus release in the restoration of very shallow lakes (The Norfolk Broads, England) and implications for biomanipulation, Hydrobiologia, 94 (1994), 445-456. doi: 10.1007/978-94-017-2460-9_39.

[31]

L. RossoJ. R. Lobry and J. P. Flandrois, An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model, J. Theor. Biol., 162 (1993), 447-463. doi: 10.1006/jtbi.1993.1099.

[32]

J. B. ShuklaA. K. Misra and P. Chandra, Modeling and analysis of the algal bloom in a lake caused by discharge of nutrients, App. Math. Comput., 196 (2008), 782-790. doi: 10.1016/j.amc.2007.07.010.

[33]

S. J. ThackerayI. D. Jones and S. C. Maberly, Long-term change in the phenology of spring phytoplankton: species-specific responses to nutrient enrichement and climatic changes, J. Ecol., 96 (2008), 523-535. doi: 10.1111/j.1365-2745.2008.01355.x.

[34]

D. ToroM. DominicD. J. O'Connor and R. V. Thomann, A dynamic model of the phytoplankton population in the Sacramento San Joaquin Delta, Adv. Chem. Ser, 106 (1971), 131-180.

[35]

C. L. WangW. Y. PanY. Q. Han and X. Qian, Effect of global climate change on cyanobacteria bloom in taihu lake, China Environmental Science, 30 (2010), 822-828.

[36]

F. B. WangS. B. Hsu and W. D. Wang, Dynamics of harmful algae with seasonal temperature variations in the cove-main lake, Discrete and Continuous Dynams. Systems -B, 21 (2016), 313-315. doi: 10.3934/dcdsb.2016.21.313.

[37]

W. D. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.

[38]

M. Winder and D. E. Schindler, Climate change uncouples trophic interactions in an aquatic ecosystem, Ecology, 85 (2004), 2100-2106. doi: 10.1890/04-0151.

[39]

M. Winder and J. E. Cloern, The annual cycles of phytoplankton biomass, Philos. T. R. Soc. B, 365 (2010), 3215-3226. doi: 10.1098/rstb.2010.0125.

[40]

Q. J. XuB. Q. QinW. M. ChenY. W. Chen and G. Gao, Ecological simulation of algae growth in Taihu Lake, J. Lake Sci., 2 (2001), 149-157.

[41]

F. Zhang and X. Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.

[42]

Q. Zhao and X. Lu, Parameter estimation in a three-dimensional marine ecosystem model using the adjoint technique, J. Marine Syst., 74 (2008), 443-452. doi: 10.1016/j.jmarsys.2008.03.006.

[43]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

show all references

References:
[1]

G. Ahlgren, Temperature functions in biology and their application to algal growth constants, Oikos, 49 (1987), 177-190. doi: 10.2307/3566025.

[2]

O. Bernard and B. Remond, Validation of a simple model accounting for light and temperature effect on microalgal growth, Bioresource Technology, 123 (2012), 520-527. doi: 10.1016/j.biortech.2012.07.022.

[3]

R. BouterfasM. Belkoura and A. Dauta, Light and temperature effects on the growth rate of three freshwater [2pt] algae isolated from a eutrophic lake, Hydrobiologia, 489 (2002), 207-217.

[4]

C. ButterwickS.I. Heaney and J. F. Talling, Diversity in the influence of temperature on the growth rates of freshwater algae, and its ecological relevance, Freshwater Biol, 50 (2005), 291-300. doi: 10.1111/j.1365-2427.2004.01317.x.

[5]

M. ChenM. FanR. LiuX. Yuan and H. P. Zhu, The dynamics of temperature and light on the growth of phytoplankton, J. Theor. Biol., 385 (2015), 8-19. doi: 10.1016/j.jtbi.2015.07.039.

[6]

W. Chen and A. Nauwerck, A note on composition and feeding of the crustacean zooplankton of Lake Taihu, Jiangsu Province, China, Limnologica, 26 (1996), 275-280.

[7]

Y. W. ChenB. Q. QinK. Teubner and M. T. Dokulil, Long-term dynamics of phytoplankton assemblages: Microcystis-domination in Lake Taihu, a large shallow lake in China, J. Plankton Res., 25 (2003), 445-453. doi: 10.1093/plankt/25.4.445.

[8]

L. DingY. Pang and L. Li, Simulation study on algal dynamics under different hydrodynamic conditions, Acta Ecologica Sinica, 25 (2005), 1863-1868.

[9]

X. H. DongH. BennionR. BattarbeeX. D. Yang and E. F. Liu, Tracking eutrophication in Taihu Lake using the diatom record: potential and problems, J. Paleolimnol, 40 (2008), 413-429. doi: 10.1007/s10933-007-9170-6.

[10]

P. Driessche and W. James, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[11]

J. J. ElserK. AcharyaM. KyleJ. CotnerW. MakinoT. MarkowT. WattsS. HobbieW. FaganJ. SchadeJ. Hood and R. W. Sterner, Growth rate-stoichiometry couplings in diverse biota, Ecology Letters, 6 (2003), 936-943. doi: 10.1046/j.1461-0248.2003.00518.x.

[12]

M. FanQ. Wang and X. Zou, Dynamics of a non-autonomous ratio-dependent predator-prey system, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 133 (2003), 97-118. doi: 10.1017/S0308210500002304.

[13]

P. J. S. Franks, NPZ models of plankton dynamics: Their construction, coupling to physics, and application, J. Oceanog, 58 (2002), 379-387.

[14]

J. A. FreundS. MieruchB. ScholzeK. Wiltshire and U. Feudel, Bloom dynamics in a seasonally forced phytoplankton zooooplankton model: trigger mechanisms and timing effects, Ecol. Complex, 3 (2006), 129-139.

[15]

G. F. FussmannS. P. EllnerK. W. Shertzer and N. G. Hairston, Crossing the hopf bifurcation in a live predator-prey system, Science, 290 (2000), 1358-1360. doi: 10.1126/science.290.5495.1358.

[16]

R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.

[17]

R. J. GeiderH. L. Maclntyre and T. M. Kana, A dynamic regulatory model of phytoplanktonic acclimation to light, nutrients, and temperature, Limnol. Oceanogr, 43 (1998), 679-694. doi: 10.4319/lo.1998.43.4.0679.

[18]

P. M. Glibert, Eutrophication and Harmful Algal Blooms: A Complex Global Issue, Examples from the Arabian Seas including Kuwait Bay, and an Introduction to the Global Ecology and Oceanography of Harmful Algal Blooms (GEOHAB) Programme, Int. J. Oceans and Oceanography, 2 (2007), 157-169.

[19]

J. C. Goldman and J. C. Edward, A kinetic approach to the effect of temperature on algal growth, Limnol. Oceanogr., 19 (1974), 756-766.

[20]

G. M. GrimaudV. L. GuennecS. D. AyataF. MariretA. Schiandra and O. Bernard, Modelling the effect of temperature on phytoplankton growth across the global ocean, IFAC-PapersOnLine, 48 (2015), 228-233. doi: 10.1016/j.ifacol.2015.05.059.

[21]

J. P. Grover and T. H. Chrzanowski, Seasonal dynamics of phytoplankton in two warm temperate reservoirs: association of taxonomic composition with temperature, J. Plankt. Res, 28 (2006), 1-17. doi: 10.1093/plankt/fbi095.

[22]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 5-60. doi: 10.4039/entm9745fv.

[23]

A. HuppertB. BlasiusR. Olinky and L. Stone, A model for seasonal phytoplankton blooms, J. Theor. Biol., 236 (2005), 276-290. doi: 10.1016/j.jtbi.2005.03.012.

[24]

K. S. JohnsonF. P. Chavez and G. E. Friederich, Continental-shelf sediment as a primary source of iron for coastal phytoplankton, Nature, 398 (1999), 697-700.

[25]

R. I. Jones, The importance of temperature conditioning to the respiration of natural phytoplankton communities, British Phycological Journal, 12 (2007), 277-285. doi: 10.1080/00071617700650291.

[26]

S. E. Jorgensen and G. Bendoricchio, Fundamentals of Ecological Modelling, Elsevier, 2001.

[27]

I. LoladzeY. Kuang and J. J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bulletin of Mathematical Biology, 62 (2000), 1137-1162. doi: 10.1006/bulm.2000.0201.

[28]

J. H. Luo, Phytoplankton-zooplankton dynamics in periodic environments taking into account eutrophication, Math. Biosci., 245 (2013), 126-136. doi: 10.1016/j.mbs.2013.06.002.

[29]

J. R. MoisanT. A. Moisan and M. R. Abbott, Modelling the effect of temperature on the maximum growth rates of phytoplankton populations, Ecol. Model., 153 (2002), 197-215. doi: 10.1016/S0304-3800(02)00008-X.

[30]

G. PhillipsR. JacksonC. Bennett and A. Chilvers, The importance of sediment phosphorus release in the restoration of very shallow lakes (The Norfolk Broads, England) and implications for biomanipulation, Hydrobiologia, 94 (1994), 445-456. doi: 10.1007/978-94-017-2460-9_39.

[31]

L. RossoJ. R. Lobry and J. P. Flandrois, An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model, J. Theor. Biol., 162 (1993), 447-463. doi: 10.1006/jtbi.1993.1099.

[32]

J. B. ShuklaA. K. Misra and P. Chandra, Modeling and analysis of the algal bloom in a lake caused by discharge of nutrients, App. Math. Comput., 196 (2008), 782-790. doi: 10.1016/j.amc.2007.07.010.

[33]

S. J. ThackerayI. D. Jones and S. C. Maberly, Long-term change in the phenology of spring phytoplankton: species-specific responses to nutrient enrichement and climatic changes, J. Ecol., 96 (2008), 523-535. doi: 10.1111/j.1365-2745.2008.01355.x.

[34]

D. ToroM. DominicD. J. O'Connor and R. V. Thomann, A dynamic model of the phytoplankton population in the Sacramento San Joaquin Delta, Adv. Chem. Ser, 106 (1971), 131-180.

[35]

C. L. WangW. Y. PanY. Q. Han and X. Qian, Effect of global climate change on cyanobacteria bloom in taihu lake, China Environmental Science, 30 (2010), 822-828.

[36]

F. B. WangS. B. Hsu and W. D. Wang, Dynamics of harmful algae with seasonal temperature variations in the cove-main lake, Discrete and Continuous Dynams. Systems -B, 21 (2016), 313-315. doi: 10.3934/dcdsb.2016.21.313.

[37]

W. D. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.

[38]

M. Winder and D. E. Schindler, Climate change uncouples trophic interactions in an aquatic ecosystem, Ecology, 85 (2004), 2100-2106. doi: 10.1890/04-0151.

[39]

M. Winder and J. E. Cloern, The annual cycles of phytoplankton biomass, Philos. T. R. Soc. B, 365 (2010), 3215-3226. doi: 10.1098/rstb.2010.0125.

[40]

Q. J. XuB. Q. QinW. M. ChenY. W. Chen and G. Gao, Ecological simulation of algae growth in Taihu Lake, J. Lake Sci., 2 (2001), 149-157.

[41]

F. Zhang and X. Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.

[42]

Q. Zhao and X. Lu, Parameter estimation in a three-dimensional marine ecosystem model using the adjoint technique, J. Marine Syst., 74 (2008), 443-452. doi: 10.1016/j.jmarsys.2008.03.006.

[43]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

Figure 2.  Seasonal patterns of water temperature at monitor sites $T_i(i = 1,2,...,12)$ from March 2008 to April 2009
Figure 3.  Seasonal patterns of Chlorophyll-a at monitor sites $T_i(i = 1,2,...,12)$ from June 2008 to April 2009
Figure 1.  Location of Lake Tai and monitor stations. The longitude and latitude of each monitor station are listed in Table 1
Figure 4.  Thermal growth rate curve and CTMI model calibrated for data of A. Formosa. (Reproduced from [4])
Figure 5.  Time-series plot of phytoplankton (solid line in red) and limiting nutrient (dash line in black) in (6). It reveals that the conditions in Theorem 3.7 and Theorem 3.11 are sufficient for the cyclic dynamics and those two theorems can be further improved. (a) $R_0<1$ and the boundary $\omega$-periodic solution is GAS (Theorems 3.7 is valid), here $T(t)=5-5\sin(\pi/6t+8)$, $P_{in1}=0.05$. (b) $R_0>1$ and the internal $\omega$-periodic solution is GAS (Theorem 3.11 holds) here $T(t)=15-15\sin(8\pi t+8)$, $P_{in1}=5$, $\mu_{opt}=5$. (c) The conditions in Theorem 3.11 are not satisfied (i.e., $R_1<1<R_0$), but (6) still admits a positive periodic solution being GAS, here $T(t)=15-15\sin(\pi/6t+8)$, $P_{in1}=0.5$. All the other parameters take the default values listed in Table 2
Figure 6.  Bifurcation diagram for (6) with $R_0$ being the bifurcation parameter. $R_0=1$ is the threshold that characterizes the global dynamics of (6), that is, the boundary $\omega$-periodic solution is GAS (phytoplankton tends to perish) if $R_0<1$ and the internal $\omega$-periodic solution is GAS (phytoplankton persists) if $R_0>1$
Figure 7.  (a) Time-series plot of water temperature $T=T(t)$. (b) Time-series plot of phytoplankton ($A(t)$) (solid line in red) and limiting nutrient ($P(t)$) (dotted line in black) in the reference model (6). It reveals that the annual cycle of temperature is the primary driver of the biomass variability. Here $T=T(t)=15-15\sin(\pi/6t+8)$ and other parameters take the default values in Table 2
Figure 8.  Solid lines show the bifurcation diagram of the minimum and maximum of periodic oscillation of phytoplankton in (6) with respect to the strength or intensity ($\alpha$) of external temperature forcing, which reveals a strong correlation between the water temperature and the cyclic dynamics of phytoplankton. Dashed line represents the bifurcation diagram for the amplitude (distance between two solid lines) of phytoplankton's periodic oscillation with respect to $\alpha$. Here $T(t)=15-\alpha\sin(\pi/6\cdot t+8)$
Figure 9.  Cyclic fluctuation pattern of monthly mean temperature of Lake Taihu (from [35])
Figure 10.  (a) Seasonal patterns of water temperature at monitor site $T_7$. In the figures, dot-solid curve represents field data while solid curve comes from fitting. (b) Comparison of model predictions (solid line) with real experimental data (stars) for the monitor station $T_7$, which shows the fundamental agreement of oscillatory population behavior and cycle phases between experiment and model prediction of Chlorophyll-a in Lake Tai
Figure 11.  (a) Seasonal patterns of mean water temperature among 12 stations in Lake Tai. In the figures, dot-solid curve represents field data while solid curve comes from fitting. (b) Comparison of model predictions (solid line) with real experimental data (stars) for the average chlorophyll-a of 12 monitor stations, which shows the fundamental agreement of oscillatory population behavior and cycle phases between experiment and model prediction of Chlorophyll-a in Lake Tai
Figure 12.  Time-series plot of phytoplankton ($A$) (solid line in black) and water temperature ($T$) (dashed line in green). The highest temperature occurs in the 5th day. The parameters are set as follows, $T_{opt}=30^oC$, (a) $T_{mean}=15$, $T_{high}=25$, $\delta=2$; (b) $T_{mean}=15$, $T_{high}=25$, $\delta=1$; (c) $T_{mean}=15$, $T_{high}=20$, $\delta=2$; (d) $T_{mean}=5$, $T_{high}=15$, $\delta=2$
Figure 13.  Time-series plot of phytoplankton ($A$) (solid line in black) and water temperature ($T$) (dashed line in green). Phytoplankton with optimal temperature $T_{opt}=23^oC$. Other parameters are same with (a) in Fig. 12
Table 1.  Location of monitor stations (MS)
MSNameLongitudeLatitude
$T_1$Jia Xing Canal${\rm E120°43'3''}$${\rm N30°48'5''}$
$T_2$Wang Jiang Jing${\rm E120°42'31''}$${\rm N30°53'6''}$
$T_3$Ping Wang Bridge${\rm E120°38'14''}$${\rm N30°59'49''}$
$T_4$Hu Zhou Source${\rm E120°4'23''}$${\rm N30°51'51''}$
$T_5$Xiao Mei Kou${\rm E120°6'1''}$${\rm N30°55'43''}$
$T_6$New Port${\rm E120°7'32''}$${\rm N30°56'18''}$
$T_7$Yi Xing Industry${\rm E119°47'48''}$${\rm N31°21'31''}$
$T_8$Zhou Tie Sewage${\rm E119°59'54''}$${\rm N31°27'20''}$
$T_9$Yi Xing Rv/Lake${\rm E120°0'30''}$${\rm N31°27'10''}$
$T_{10}$Wu Xi Mei Liang${\rm E120°7'25''}$${\rm N31°30'13''}$
$T_{11}$Wu Xi Lake${\rm E120°21'4''}$${\rm N31°27'57''}$
$T_{12}$Su Zhou Lake Tai${\rm E120°46'55''}$${\rm N31°13'17''}$
MSNameLongitudeLatitude
$T_1$Jia Xing Canal${\rm E120°43'3''}$${\rm N30°48'5''}$
$T_2$Wang Jiang Jing${\rm E120°42'31''}$${\rm N30°53'6''}$
$T_3$Ping Wang Bridge${\rm E120°38'14''}$${\rm N30°59'49''}$
$T_4$Hu Zhou Source${\rm E120°4'23''}$${\rm N30°51'51''}$
$T_5$Xiao Mei Kou${\rm E120°6'1''}$${\rm N30°55'43''}$
$T_6$New Port${\rm E120°7'32''}$${\rm N30°56'18''}$
$T_7$Yi Xing Industry${\rm E119°47'48''}$${\rm N31°21'31''}$
$T_8$Zhou Tie Sewage${\rm E119°59'54''}$${\rm N31°27'20''}$
$T_9$Yi Xing Rv/Lake${\rm E120°0'30''}$${\rm N31°27'10''}$
$T_{10}$Wu Xi Mei Liang${\rm E120°7'25''}$${\rm N31°30'13''}$
$T_{11}$Wu Xi Lake${\rm E120°21'4''}$${\rm N31°27'57''}$
$T_{12}$Su Zhou Lake Tai${\rm E120°46'55''}$${\rm N31°13'17''}$
Table 3.  Values used for field application
MS $T_{mean}$ $\alpha$ $\varphi$ $P_{in1}$ (mg$\cdot$L$^{-1}$)
$T_7$19.18-11.377.15270.364
$Mean$18.90-11.680.86110.242
MS $T_{mean}$ $\alpha$ $\varphi$ $P_{in1}$ (mg$\cdot$L$^{-1}$)
$T_7$19.18-11.377.15270.364
$Mean$18.90-11.680.86110.242
Table 2.  Parameters of model (6) with default values used for numerical studies
Par.DescriptionValueUnitRef.
$U_p$maximum nutrient uptake coefficient $0.05$ $\mathrm{day}^{-1}$[40]
$\mu_{opt}$maximum growth rate of phytoplankton $0.3-1.6$ $\mathrm{day}^{-1}$[2]
$T_{opt}$optimal water temperature $20-30$ $^{\circ}\mathrm{C}$[2]
$T_{min}$minimum water temperature $-10-0$ $^{\circ}\mathrm{C}$[2]
$T_{max}$maximum water temperature $30-40$ $^{\circ}\mathrm{C}$[2]
$T_{ref}$reference water temperature $20$ $^{\circ}\mathrm{C}$[8]
$m_0$natural mortality rate of phytoplankton $0.13$ $\mathrm{day}^{-1}$[8]
$K_1$Respiration rate of phytoplankton $0.1-0.5$ $\mathrm{day}^{-1}$[8]
$\theta_1$temperature dependence coefficient $1.11$ $-$[8]
$K_2$temperature dependence coefficient $0.0004-0.1$ $\mathrm{day}^{-1}$[26]
$\theta_2$temperature dependence coefficient $1.02-1.14$ $-$[26]
$H_1$coefficient for nutrient uptake $0.6227$ $\mathrm{mg}^{2}\cdot \mathrm{L}^{-2}$Defaulted
$H_2$coefficient for nutrient uptake $0.0320$ $\mathrm{mg}^{}\cdot \mathrm{L}^{-1}$Defaulted
$r_1$coefficient for nutrient uptake $2$ $-$Defaulted
$r_2$coefficient for nutrient uptake $1$ $-$Defaulted
$P_{in1}$density of total input soluble phosphorus $0.364$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$Defaulted
$P_{in2}$density of total input solid phosphorus $0.5$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$Defaulted
$P_{b}$density of solid phosphorus in bottom sediments $0.5$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$Defaulted
$a$dilution rate $0.62$ $\mathrm{day}^{-1}$Defaulted
$b$density limiting coefficient $0.023$ $\mathrm{L}\cdot(\mathrm{day}\cdot\mathrm{mg})^{-1}$Defaulted
Par.DescriptionValueUnitRef.
$U_p$maximum nutrient uptake coefficient $0.05$ $\mathrm{day}^{-1}$[40]
$\mu_{opt}$maximum growth rate of phytoplankton $0.3-1.6$ $\mathrm{day}^{-1}$[2]
$T_{opt}$optimal water temperature $20-30$ $^{\circ}\mathrm{C}$[2]
$T_{min}$minimum water temperature $-10-0$ $^{\circ}\mathrm{C}$[2]
$T_{max}$maximum water temperature $30-40$ $^{\circ}\mathrm{C}$[2]
$T_{ref}$reference water temperature $20$ $^{\circ}\mathrm{C}$[8]
$m_0$natural mortality rate of phytoplankton $0.13$ $\mathrm{day}^{-1}$[8]
$K_1$Respiration rate of phytoplankton $0.1-0.5$ $\mathrm{day}^{-1}$[8]
$\theta_1$temperature dependence coefficient $1.11$ $-$[8]
$K_2$temperature dependence coefficient $0.0004-0.1$ $\mathrm{day}^{-1}$[26]
$\theta_2$temperature dependence coefficient $1.02-1.14$ $-$[26]
$H_1$coefficient for nutrient uptake $0.6227$ $\mathrm{mg}^{2}\cdot \mathrm{L}^{-2}$Defaulted
$H_2$coefficient for nutrient uptake $0.0320$ $\mathrm{mg}^{}\cdot \mathrm{L}^{-1}$Defaulted
$r_1$coefficient for nutrient uptake $2$ $-$Defaulted
$r_2$coefficient for nutrient uptake $1$ $-$Defaulted
$P_{in1}$density of total input soluble phosphorus $0.364$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$Defaulted
$P_{in2}$density of total input solid phosphorus $0.5$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$Defaulted
$P_{b}$density of solid phosphorus in bottom sediments $0.5$ $\mathrm{mg}\cdot\mathrm{L}^{-1}$Defaulted
$a$dilution rate $0.62$ $\mathrm{day}^{-1}$Defaulted
$b$density limiting coefficient $0.023$ $\mathrm{L}\cdot(\mathrm{day}\cdot\mathrm{mg})^{-1}$Defaulted
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